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1. Harmonic Loading The loading consists of one or more load cases each of which may have a phase angle different from zero f load frequency Q rad s amp f 2 Hz Lo jEva Harmonic and periodic loading Harmonic loading Periodic loading Harmonic loading One load frequency Several load frequencies 4 Load frequency increment Af Snax I Pee La A Od SA SND PE E fi 5 Jaat n frequency increments 7 in load frequency range From frequency f_1 0 00 Hz to frequencey f_max 0 00 Hz Number of load frequencies in range n 0 Specify the spatial load combination which is the same for each load frequency Press here gt f Hz Harmonic and periodic loading ic loading Hal Periodic loading defining the time variation of one load period This loading is approximated by a Fourier series Specify LCmb Press here z Periodic load and time function Press here a Number of time increments in period 0 Maximum number of Fourier terms 75 Tolerance for Fourier approximation 0 050 A periodic loading is defined by one specified spatial load combination LCmb and one specified time function Fov Analysis and results a Harmonic loading with one load frequency b Harmonic loading with several load frequencies c Periodic non harmonic loading Figure 47 Load options in the frequency domain Page 42 fap2D Analysis a
2. Linear static Steel design Figure 22 Results available for a linear static analysis Page 25 fap2D Analysis and results ments and section forces N M and V These can be shown all four at once Show all or one diagram at a time Figure 23 shows the bending moment diagram It 2 Figure 23 Bending moment diagram and detail results should be noted that this diagram is always drawn on the tension side of the member The only result accompanying the diagram is the maximum moment value 26 44 kNm not shown in figure 23 However by left and right clicking an element results at the element s ends are shown in a result box as shown in figure 23 Clicking the o t button in this box will as shown produce another box with the axial stress 0 at the extreme cross section fibers as well as the maximum shear stress t The latter is not available for cross sections of the arbitrary category All individual diagrams can be scaled up or down by using the arrow buttons in the toolbox they can be normalized again using the N button For the displacement dia gram the toolbox provides a button T8 that will show the displacements with true real size To the left in the ribbon figure 22 are two more buttons Reaction forces and Resul tants Clicking the first will produce a view of the model with arrows indicating all non zero reactions right clicking the joint symbol will produce a box with the
3. 1 2 nme foe A WINDOWS based program for static and dynamic analysis of 2D frame type structures USER s MANUAL Kolbein Bell May 2011 Content Preface Capabilities The structural model Members Joints Boundary conditions Eccentricities Spatial loading Time dependent loading Frequency dependent loading Periodic none harmonic loading Mass Damping The computational model Basic philosophy Reference and identification Elements Solution Modelling structure and loading GUI basics Modelling the structure Modelling the loading Analysis and results Linear static analysis Computational aspects Typical results Linearized buckling analysis Computational aspects Typical results Nonlinear static analysis Computational aspects 10 11 11 13 14 15 15 16 17 17 18 19 20 21 21 22 23 25 25 25 25 2 2 28 29 30 Page 2 fap2D Typical results Free undamped vibration analysis Computational aspects Results Forced vibration analysis time domain Computational aspects Typical results Forced vibration analysis frequency domain Computational aspects Typical results Influence line analysis and extreme response Computational aspects Typical results Steel design 31 34 34 35 35 38 38 41 43 44 47 48 48 50 Page 3 Preface The development of program fap2D started at the Department of structural engineering at NTNU Norwegian University
4. KG P 2 Page 27 fap2D Analysis and results where K is the material stiffness and Kg which is a function of the axial forces P is the geometric stiffness The minus sign in equation 2 assumes the axial forces taken positive as compression which is a common convention in buckling analysis The material stiffness is identical to the ordinary 1st order stiffness K modified with respect to bi linear bar elements Shear deformations may be included in K The geometric stiffness matrix may be expressed as Ks Ke pK 3 where Kg is the geometric stiffness due to the axial forces P caused by R acting alone and Kg is the geometric stiffness due to the axial forces P caused by the nominal R acting alone Hence K Ky Kg pKa Ki PKg 4 where K is the material stiffness modified with respect to the geometric stiffness effects of the constant part of the loading that is K K K 5 Buckling is now defined as a state for which K becomes singular For a singular K the homogeneous system of equations K q K pKg q 0 6 has non trivial solutions p q Equation 6 represents a general symmetric eigen problem This problem is by default solved by so called subspace iteration However in the lower right hand corner of the Run analysis button the small arrow will launch a dialog box from which it is possible to choose a truncated Lanczos method for the eigenvalue extraction The se
5. NOTE Arch members cannot be divided into ordinary members NOTE Both beam and arch members can be divided into sub members by Page 8 fap2D The structural model introducing internal joints However the sub member remains an integral part of its mother member and the only properties it can have that differ from those of the mother member are associated with distributed loading Figure 3 shows arch members of circular and parabolic type Regardless of how they are created they are uniquely defined by the coordinates of the base points A and B and the radius of curvature R and height A respectively As explained below in the circular arch member parabolic arch member Figure 3 Arch members joint section arch members are somewhat sensitive to changes made to the position of base points A and B and or R and h Joints The following rules apply 1 Ifa joint is deleted all members connected to it are automatically deleted However the opposite does not apply if all members connected to a joint are deleted the joint is not deleted It follows from this that joints take precedence over members i e members are connected to joints not the other way around 2 An internal joint resides at the interior of a host member If an internal joint is deleted all members connected to it except its host member are deleted 3 The position of a joint may be changed arbitrarily For an arch member connected to a joint th
6. moving a support A dependant dof is defined by the simple constraint equation Page 10 fap2D The structural model where the dependent or slave dof r is set equal to a master dof r which must be a free dof This simple slave concept is the tool provided for modelling all types of hinges or displacement releases However the details of this are all well hidden for the user The constraint equations required to handle a displacement release are auto matically created by the program once the user has defined in a fairly intuitive way the kind of release he or she wishes to impose It should be noted that the degrees of freedom at a joint follow the coordinate axes at the joint This will be the global axes unless the user has defined local axes at the joint which she he can do at any joint of the model Suppressed prescribed and dependent dofs are collectively referred to as specified dofs By this terminology we have two types of dofs free or unknown dofs and specified or known dofs Instead of or in combination with specified dofs elastic springs may be used to simu late boundary constraints One example of effective combination is the modelling of a semi rigid joint A coupling spring may be attached to the degrees of freedom created at a joint through a releasing hinge Eccentricities Eccentricities and very stiff parts of a structure are most conveniently and safely modelled as completely rigi
7. o cl y i a Fa Ws sin a hy 20dy 2 N In the figure above A is a negative number If 4 gt 0 then y 0 for lt g All A values are measured in terms of time units Type 5 Sine function B ee A A i w pumai a ua an o a A 7 i gt lt lt per a Ap 7 hv t i a Mp pla Ap Types 1 and 2 are uniquely defined by their type numbers they require no other information than a er Type 3 requires n pairs of numbers A and y plus the number of periods n whereas types 4 and 5 require Ay A A and nper Frequency dependent loading AR gt eriod T gt P Q f f 3 is the frequency in Hz Figure 8 Harmonic loading Frequency dependent loading is harmonic with a variation along the time axis as Page 14 fap2D The structural model shown in figure 8 The load frequency is Q rad s or f Hz fam2D recognizes two types of harmonic loading 1 Only one load frequency Q In other words all loading is harmonic in time and have the same frequency However more than one spatial load combination may be applied and these may have different phase angles a In fact if the phase angles are not different it makes little sense to apply more than one spatial load combination and all spatial loading may as well be lumped into one load combi nation 2 Only one spatial load combination is
8. B 0 575 values determined from a gives an unconditional y 0 2889 stable algorithm PoJ pva Figure 42 Numerical integration Typical results Edit cross section Name Preview I cross section Cross Section Parameters bt mm 2 n E mm 4 Dimensions tt mm b h 300 00 mm 7 j b beam cross section bt 200 00 mm mm bb 200 00 mm tb __ tw 15 00 mm tt 20 00 mm _bb_ tb 20 00 mm Figure 43 Example problem Page 38 fap2D Analysis and results Before pushing the Run analysis button make sure that at least one response parameter has been defined In order to demonstrate the capabilities of this type of analysis we consider a very simple problem namely the simply supported steel beam shown in figure 43a The beam has a parametric cross section shown in figure 43 b and it is subjected to a vertical point load of 10 KN at the middle of the beam The time vari Edit time function Name Pulse Details Time s Amplitude et o gt BR v Zoom contents Number of periods 10 Py aro Figure 44 Time function Pulse ation of the load is a pulse of 1 0 sec duration shown in figure 44 Ten consecutive periods of this pulse is considered as the total time function Back to the time axis parameters see figure 41 we now somewhat arbitrarily specity the time increment to be AA 0 001 s We sample the response parameter the vertical di
9. Concentrated loads including moments Concentrated point loads in the direction of the global reference axes can be applied anywhere within beam and arch members but only at member ends for bar cable and strut members Concentrated moments can only be applied to beam and arch members Concentrated point loads can only be applied at joints or internal joints If a point load is applied inside a beam or arch member where there is no internal joint such a joint will be automatically inserted by the program NOTE All external loading is conservative 3 Initial strain e g temperature can be applied to any member For each member an initial strain condition that is constant in the element direction but may have a linear variation over the member height may be specified see figure 7 4 Prescribed displacements also have a load effect However prescribed displace ments are treated as boundary conditions and as such are dealt with in the follow ing section Each external load whether distributed or concentrated must be assigned to a named load case LC The user may define any number of distinctly named load cases y and any one load case can contain any number of indi y vidual loads Two LC s are created automatically one called Default load case and one called Own weight The first will accommodate all external loading not specifi cally assigned to another named LC whereas the lat
10. E eccentricities rigid links at point C or response parameters right click the joint and select the appropriate function from the popup menu which in turn will lead you to a fairly self explanatory dialog box Modelling the loading w Modelling Loading Analysis Results Settings Show all loads LC ty f e amp T N LCmb dd m dd i bed Add d i sine load load displacements strain temp functions Show selected LCmb Load case 4 Spatial loading direct and indirect Time functions Load combination s View Loading Figure 18 The loading ribbon Also the loading ribbon shown in figure 18 is fairly self explanatory spatial loading in terms of load cases LC and time functions are defined here It should be noted that prescribed displacements and initial strain both gives rise to spatial loading All spa tial loading must be assigned to a particular LC Two predefined load cases exist Default load case and Own weight and the user may define any number of named load cases via the Add edit command in the drop down menu Own weight contains the dead load of the structural members themselves and this loading is automatically computed by the program other loading cannot be assigned to this LC The Default load case and any user defined LC can accommodate any number of concentrated loads moments and or distributed member loads If prescribed displacements are defined at one or more supported joints the program automati
11. a simple sensitivity analysis in which results Page 39 fap2D Analysis and results Response parameters Current response parameter w z displacement at joint 2 Full MDOF model Response mm Response parameters Current response parameter w z displacement at joint 2 ao AAA AVL Modal analysis with ar n e E the 10 lowest free i rr S a vibration modes as modal cooordinates Response mm Response parameters Current response parameter w z displacement at joint 2 ai i a i 10 load dependent o M E Ritz vectors as modal cooordinates Response mm Figure 45 Time history plots of the vertical displacement at the mid point of the beam obtained with different values of the time increment AA are compared However with the current computational power of a standard PC some overkill is probably justified Page 40 fap2D Analysis and results Forced vibration analysis frequency domain Figure 37 shows the ribbon for forced vibration analysis in the time domain The Run ap f D Programs Frame2D Examples Beam_dyn fap2D fi Modelling Loading Analysis Results Appearance H i bot Ee Ay 4 a A pas al Cm ip x K fh a 4 z Linear Non linear Buckling Free vibration Dynamic Dynamic static static time domain frequency domain Influence Include Computional Stiffn
12. applied as a harmonic load but it may be applied with many different frequencies from Q to Qax in equal steps of AQ In this case phase angle has no meaning for the loading max Periodic but non harmonic loading iv load multiplier A time function of type 3 4 or 5 Ps r y gt t A period T A p T p T p Figure 9 Periodic loading This loading is characterized by one or more spatial load combinations and a time function vy the same time function for all contributing loading which can be due to external loading and prescribed displacements The time function must be of type 3 4 or 5 see above section about time dependent loading The periodic loading exemplified in figure 9 is automatically replaced by a series of harmonic load combinations through a Fourier series analysis The number of Fourier terms included is determined by a user defined tolerance parameter subject to a ceiling defined by the maximum permissible number of terms which is also speci fied by the user Mass The mass of the structural members may be automatically accounted for by one of three mass representation models e lumped mass representation the mass of each element is lumped into two equal concentrated translational masses at the element nodes all rotational dofs are mass less e consistent mass representation the element mass matrix is established on the basis of t
13. convenience we have assigned the loading to the Default load case It should be noted that rigid arms or links cannot take distributed loading any such loading must be transferred by the user to the joint at the end of the link as one or two forces depending on the link s orientation and a moment Add line load to member s Select load case Load data Type of load Gravity load Linear variation along member Load intensities End 1 10 00 End 2 5 00 Figure 20 A spatial LC Default load case Page 24 Analysis and results Linear static analysis Figure 21 shows the ribbon for linear static analysis The Run analysis arrow is active and if pressed the analysis will be carried out without shear deformations for the Default load combination Shear deformations are either neglected which is default or included by pressing the Include shear deformation button this button toggles off on The LCmb drop down menu enables the user to select any existing load com bination for the analysis in fact the user can also define new load combinations or edit existing ones from this position the last item in the drop down menu is Add edit but only using existing load cases LC If new LCs are required it is necessary to go back to the Loading ribbon a P Modelling Loading Analysis Results Settings i d o aii ea at Ha ip LCmb 7 inear Non linear Bucklin
14. least some of the buttons As for all other analyses the Include shear deformation button is optional default is no shear deformations The Computational model button is also optional in that a viable default setting is in place This button launches the dialog box in figure 38a which shows the default setting A full and coupled multi degree of freedom MDOF model is default The alternative is modal analysis leading to a series of decoupled single degree of freedom equations SDOF If modal analysis is chosen we have a choice of modal coordinates free vibration mode shapes which are default see figure 38b or load dependent Ritz vectors If the latter is chosen a spatial Page 35 fap2D Analysis and results Computional model Computional model D Full coupled MDOF model Modal analysis uncoupled SDOF model Modal analysis uncoupled SDOF model Modal analysis Number of modes 10 Modal coordinates are Eigenvectors corresponding to eigenvalues closest to the shifted origin Load dependent Ritz vectors based on load combination specified belov Shift value 0 Hz Load combination Note Care should be exercised when changing the shift value shifting the origin of the characteristic polynomial is a highly problem dependent operation Le jtv 9 ji v a default setting b modal analysis Figure 38 Computational model for forced vibration analysis in the time domain
15. placed at a nodal point of a member of the load path The number of load situations i e the number of load vectors is therefore defined by the total number of nodal points in the load path members Since the model is linear the stiffness matrix is formed and factorized once and for each load situation the solution is obtained by forward and backward substitutions which are cheap operations compared with factorization For each load situation the response parameters are computed and stored for presentation For a particular response parameter each such computed value is drawn as a scaled line segment perpendicular to the horizontal vertical projection ot the travel path starting from the x z coordinate of the node in question The influence line of this particular response parameter is the line drawn through the end points of these scaled lines Typical results As a simple example we consider the continuous beam in figure 53 and we seek the influence lines for the vertical displacement at joint 2 the reaction force at joint 4 and the bending moment over the support at joint 5 The load path is the entire beam grid spacing 0 5 m RHS 150x100x8 Figure 53 Example problem for influence line analysis which is 18 m long indicated by the green color in the figure The influence lines are shown in figure 54a b and c The maximum values indicated are those caused by the most critical position of the unit point
16. values of the reaction forces Right clicking any joint will produce the residual forces at the joint for an unsupported joint with no displacement releases hinge these forces should be zero At a hinge the residual forces are the hinge forces the sum of which should be zero for all members at the joint The Resultants button will produce the sum of all external loading in the two global directions as well as the sum of all reaction forces in the same directions Page 26 fap2D Analysis and results Linearized buckling analysis Figure 24 shows the ribbon for linearized buckling analysis The Run analysis arrow mat D Programs Frame2D Exar Modelling Loading Analysis Results Settings i l i ae ee orn ps a pad ip 4 7 ia i Number of modes 5 Linear Non linear Buckling Free vibration Dynamic Dynamic Influence Include Load history Run Analysis static static time domain frequency domain lines Choose analysis type Shear deformation Loading Buckling modes Buckling analysis Figure 24 Analysis ribbon for linearized buckling analysis is active but unless you have made a visit to the Load history button you most likely will receive an error message telling you that the buckling analysis is aborted due to very low level of compression forces Load history launches the dialog box shown in figure 25a Here you will have to make some assumptions and then specify the load ing You can a
17. 9c As expected both amplitude and DLF peak at the lowest natural frequency 3 89 Current response parameter w Z displacement at joint 2 ks Amplitude DLF Phase angle a a Amplitude Current response parameter w z displacement at joint 2 a Amplitude Lor Phase angle A ri b DLF dynamic amplification o c Phase angle Figure 49 Transverse mid point displacement due to a harmonic point load P 10 kN Page 45 fap2D Analysis and results Hz Finally we subject the beam to a point load with a periodic but non harmonic variation in time The load acts as before at the mid point of the beam and the time variation of one period which is 1 0 s long is the pulse shown in figure 44 This is a simple example of the loading situation of figure 47c Again results will only be available for specified response parameters of which we still have defined only one the transverse displacement under the load We continue to use the program s default settings for computational model stiffness mass and damping Also for the parameters controlling the FOURIER series approximation the we use the default values see Figure 47c We determine the response at each of 200 equally spaced points in a time period of 1 0 sec The steady state response of the mid point displacement over a time period is shown in figure 50a We see that this plot compares
18. In order to satisfy the code x should be smaller than or equal to 1 one 2 The cross section control is a fairly complex control which will not be dealt with in any detail here In short the code defines 4 classes of cross sections the purpose of which is to incorporate the effect of local buckling of the cross section itself on the section s capacity in such a way as to avoid this type of buckling Controls are carried out for the section forces present individually and in combination according to the rules of the code Sections of class 1 and 2 are checked for their plastic capacity while sections of class 3 are subjected to an elastic control It should be emphasized that currently the program does not handle sections of class 4 3 The component control is applied to all straight steel beam members of constant cross section subject to compression The in plane buckling load is estimated through the results from a linearized buckling analysis carried out automatically by the program for the actual load combination For each relevant member the appropriate controls considering the axial compression and the largest bending moment are carried out and the entire member is attributed this highest index Sections not considered are assigned class number 0 and members not considered are assigned the color white in the capacity color map of the structural model Finally it should be kept in mind that the program being limited t
19. Modelling structure and loading GUI basics 96 New structural model 0 fap2D ASJ o gt 4 Modelling Loading Analysis Results Appearance Select material ae Start Spec memnegi ne z D 7 Hinge X dof Add point Add point Manage Ri b b O n release mass damper meshing Beam mem condition and rel Mass Damping Mesh Boundary Coupling Add joint Draw member sim in BC Cross section fy Member Spring Joint Bounda New structural model 0 Ribbon page Menu line User Manual Application menu Left panel Modeling panel Joint data HEEG Bottom panel 4 Print Screen coordinates X 730 00 Y 343 World coordinates X 4 42 Z 3 33 Figure 14 GUI overview Figure 14 shows an overview of the GUI The panels shown in the figure are also referred to as dock panels since they may be docked anywhere in the GUI view The first thing a new user has to do once the program provides the fap2D window is to push the application menu button at the top left corner This will launch the menu shown in figure 15 Recent structural models will at this stage be empty and the nat ural choice is therefore to push New which produce the window shown in figure 14 The program makes use of the ribbon concept and basically the user works from left to right Apart ey T from this manual a pdf version of which is available s from the question mark button at the rig
20. a linear ana lysis requires some estimate of the second order effects e g buckling load If anonlinear analysis is carried out the computed section forces already incorporate the higher order geometric effects Hence the only control available following a non linear analysis is the elastic stress control Figure 57b shows the capacity color map for the elastic stress control following a nonlinear analysis of the frame of figure 55a The frame has in addition to the loading of figure 55a been subjected to a geometric imperfection in the shape of the first buckling mode shown in figure 57a Max amplitude 12 mm Max capacity index 0 95 a Geometric imperfection b Elastic stress control Figure 57 Steel design based on nonlinear static analysis with imperfection Page 52
21. are assumed The loading which may consist of external forces initial strains e g temperature and prescribed displacements may as explained above be applied in one step or in a number of equal increments In the latter case individual response parameters may be determined sampled after each load step resulting in a response history for each parameter It should be noted that if a step wise loading procedure is chosen part of the loading may be kept constant during the loading process Hence the program provides the three loading options shown in figure 28 R and R are the load vectors loading loading loading Figure 28 Loading options for nonlinear static analysis corresponding to spatial load combinations It should be noted that the variable part of the loading R must consist of external loads and or prescribed displacements but not initial strain the constant part of the loading R may contain all types of loading If more loading than the structure can sustain is applied the subroutine will auto matically reduce and adjust the loading to a level within 1 of the maximum load ing the structure can support before it becomes unstable This needs an explanation since maximum loading is to some extent dependent on how the loading is applied If all loading is applied in one step then depending on the magnitude of the loading one of two things can happen 1 T
22. at is moved the move will have certain consequences explained below For all other members the move means change of length and or orientation 4 The position of an internal joint may be changed but only along the host member axis 5 Internal joints may be introduced in one of four ways a by placing it on the axis of an existing beam or arch member which then becomes a host member for the internal joint b by subdividing an ordinary beam or arch member c by joining anew member to a point located at the interior of an existing member the latter becomes the host member for the internal joint and finally d by applying a concentrated load to a point inside a member concen trated or point loads can only act at a joint or internal joint NOTE An internal joint can only have one host member Page 9 fap2D The structural model What happens if the shape of an arch is changed by moving one or both of its base points A and B and or by changing the parameter R or h If the member has no internal joints there is no problem The member s geometric definition is simply updated If however the member has internal joints a well defined and unique procedure for the new location of the internal joints is necessary By definition they reside on the new position of the member axis and their relative distance trom the first base point A is the same as before the change measured along the chord which is the straight line betw
23. aving specifically defined any loading for the model NOTE If prescribed displacements have been specified for suppressed dofs the Prescribed displ LC has been created An analysis carried out for a load combination that do not include this LC will assume that the dof s in question is are suppressed Time dependent loading Time defined by the time parameter A means real time in seconds if we are talking about a dynamic analysis whereas it is pseudo time in the case of a nonlinear static analysis used to define the loading and response history Time dependent loading consists of a spatial load combination multiplied by a time dependent load factor y which is referred to as a time function The program recog nizes 5 different time function types valid between 0 and pax Typei1 w 1 0 constant Type 2 y linear between 0 and 1 maxX Type 3 Arbitrary but possibly semi periodic W3 number of periods 11 r is 3 here a p N yer 1 p e r 7 _e 2 P p4 P p W is defined by n pairs A W3 of numbers and the number 7 of periods all of which are input information The A values are measured in terms of time units The function value varies linearly between the points and the requirements are Nip gt A and A X nper lt Amax per Page 13 fap2D The structural model Type 4 Sine function A Ao Ay Za
24. cally creates an LC named Prescribed displ all prescribed displacements of a particular structural model will be associated with this LC which can only hold pre scribed displacements no other loading Similarly if temperature or other types of initial strain is defined for one or more members the program automatically creates an LC named Init strain all loading of type initial strain will be associated with this LC which cannot hold any other type of loading Named time functions of types 3 4 or 5 see the section on time dependent loading on page 13 are defined via the Edit time functions button Manage spatial load combinations Load Combination Load Combinations in model New load combination E Figure 19 Dialog box for creating and defining spatial load combinations Page 23 fap2D Modelling structure and loading Spatial load combinations LCmb may be defined via the LCmb button in the Loading ribbon see figure 19 The program has one predefined load combination Default load combination which initially contains 1 0xDefault load case if Default load case contains at least one load if not Default load combination contains 1 0xOwn weight It should be noted that while spatial load combinations can also be defined in the Analysis ribbon for some analysis spatial load cases and time functions can only be defined here in the Loading ribbon Figure 20 shows a spatial LC for the frame of figure 17 for
25. ch a dialog box from which it is possible to choose a truncated Lanczos method for the eigenvalue extraction Results The only results from a free vibration analysis are the mode shapes and correspon ding frequencies in Hz Displacement plots very similar to the buckling mode shapes are produced accompanied by the corresponding frequency For the frame in figure 17 the first or lowest mode shape is very similar to the second buckling mode shown in figure 25b its frequency is 7 28 Hz A useful feature is animation of the free vibration mode shapes a start and a stop but ton is available in the toolbox for the result view Forced vibration analysis time domain Figure 37 shows the ribbon for forced vibration analysis in the time domain The Run D Programs Frame2D Examples Beam_dyn fap2D fap2D eu Modelling Loading Analysis Results Appearance fey Bey ao ip x wa E J At fry y Linear Non linear Buckling Free vibration Dynamic Dynamic Influence Include Computional Stiffnes Mass Damping Loading Time axis Numerical Run Analysis static static time domain frequency domain lines model integration Choose analysis type Shear deformation Dynamic model Stiffnes mass and damping Load option Time domain Integration Dynamic analysis time domain Figure 37 Analysis ribbon for forced vibration analysis in the time domain analysis arrow is active but in order to carry out an analysis the user needs to visit at
26. d links or arms at the end of one or more members A stiff corner of some size in a frame as indicated by figure 5a may for instance be mod elled as shown in figure 5b The red arms are completely rigid and weightless links Such links may be introduced at the end of any member in the model and they need not be mere extensions of the member In other words the rigid links may have directions that are completely independent of the member they are attached to beam beam member a Joint rigid link column a b Figure 5 A stiff frame corner a and its model b In many cases it is better to model something very stiff as completely stiff using rigid links instead of one or more very stiff members Spatial loading The following types of spatial loading may be applied 1 Distributed loading Four types of distributed loading gravity loading projection loading both horizontal and vertical normal loading Page 11 fap2D The structural model tangential loading may be applied to any member see figure 6 All distributed loads can have a linear variation along the member or its projections in the horizontal or vertical direction Furthermore a distributed load may be applied to the entire member or to one or more sub members vyvvyyyy gravity loading projection loading normal loading tangential loading dead load snow load wind load Figure 6 Distributed loading 2
27. de Stiffmes Mass Shift value Run Analysis static static time domain frequency domain lines Choose analysis type Shear deformation Stiffness and mass Frequencies and shift values Free vibration analysis Figure 34 Analysis ribbon for free vibration analysis active and if pushed a free vibration analysis will be carried out for default choices for stiffness and mass Figure 35 shows the dialog boxes behind the Stiffness and Mass buttons We see that it is possible to modify the structural stiffness by including Stiffnes The stiffness matrix of the model is K K K Mass representation where K is the standard 1st order material l Lumped mass stiffness and K is the geometric stiffness corresponding to a specified loading Diagonilzed mass Consistent mass K K_0 Material stiffnes only DK K_0 K_G Including geometric stiffnes Load combination producing geometric stiffness Figure 35 Stiffness and mass settings for free vibration analysis the geometric stiffness due to a particular load combination before carrying out the free vibration analysis see next section An example of this would be to determine the free vibration characteristics of a structure subjected to all dead load e g a bridge For structures with cable members this may make a significant difference The default choice for the mass representation is the obvious one for the modelling philos ophy
28. e which involves updating the geometry until the unbalanced forces are sufficiently small If the same total loading is applied as a variable load in say 20 equal increments the structure may well be capable of sustaining considerable more load than if the same load was applied in just one step This will be demonstrated by a simple example in the next section and it has to do with the fact that gradual loading facilitates force redistribution to take place which in turn may give to structure more apparent strength Apparent because material failure will most likely occur long before the maximum stability load is attained This discussion is therefore of more academic than practical interest Typical results The results available from a nonlinear 100 KN static analysis are the same as those obtained by a linear analysis except that a nonlinear analysis may also provide time history plots for specified response parameters The latter may give useful information about the nonlinearity of the problem The simply supported EULER column in figure 29a serves as an example A linear ized buckling analysis suggests a buck ling load of 614 KN figure 29b Next we increase the load to 1000 KN and carry out a nonlinear analysis with a geo metrical imperfection with the shape of grid spacing 0 5 m eee the first buckling mode figure 29b and an amplitude of 16 mm L 250 SHS 100x100x10 a b First we apply a
29. ed member is a host member Internal joints divides a member into sub members internal joint member host for an internal joint host member i member joint Figure 2 Internal joints and sub members Page 7 fap2D Members The structural model The following types of structural members are available Straight beam members henceforth called beam members Curved beam members henceforth called arch members arch members may have circular or parabolic shape Straight bar members axial force only both tension and compression Straight cable members axial tension only bi linear elastic behaviour Straight strut members axial compression only bi linear elastic behaviour In addition to these types of members elastic springs both boundary springs suppott ing a particular displacement component and coupling springs connecting two similar displacement components or degrees of freedom at coinciding nodes can also be applied to the structural model The cross section of a member belongs to one of the following three categories 1 pA Predefined that is standardized steel sections whose properties are tabulated Parametric that is a cross section with a geometric form rectangle circular tube etc that is uniquely defined by a set of geometric parameters and for which the necessary mechanical properties are determined by the program using the geometr
30. een A and B Both members and joints are numbered in the order they are created While internal joints are included in the joint numbering series the sub members are not numbered The numbers may be shown or hidden Depending on how the model is established the numbering can be quite erratic and it may thus prove useful to use the renum bering facility located in the toolbox Boundary conditions Boundary conditions are defined at joints All joints and internal joints will become nodal points in the computational model and as such each has three kinematic degrees of freedom two orthogonal displacements and one rotation Any one degree of freedom may be e free or unknown e suppressed which means it has a fixed value of zero e prescribed which means it has a fixed non zero value or it may be e dependant of coupled to another free degree of freedom Figure 4 shows the various possibilities of suppressing degrees of freedom at a joint along with their symbols all 3 dofs are suppressed the two translational dofs are suppressed one translational dof is suppressed TEADA A G44 one translational and the rotational dof are suppressed ta the rotational dof is suppressed Figure 4 Suppressed degrees of freedom A prescribed dof can only be imposed on a suppressed dof In other words a pre scribed dof which in most cases will result in an indirect loading effect is imposed by
31. erence and identification axial element WW nodal degrees of freedom a esi Figure 11 Typical computational model With reference to figure 11 the computational model consists of straight beam and axial elements interconnected at nodal points or just nodes Elastic spring elements may also be included The model is referred to a global or reference coordinate system Z Each nodal point is assigned a unique number ranging from to number of nodes and each element is numbered consecutively from to number of elements This latter number series includes both beam and axial bar cable strut elements in any order but spring elements are numbered in a separate series This numbering does not really concern the user furthermore the node numbering is automatically opti mized with respect to equation solving by a fairly efficient renumbering scheme Each nodal point has three kinematic degrees of freedom dofs two orthogonal trans lations and one rotation By default the dofs also denoted nodal displacements are referred to are parallel with the global reference axes x and Z It is however pos sible to define a local coordinate system x z at any node At such a node the trans lational dofs follow the local axes At a node where only axial elements meet e g node 6 in figure 11 the program auto matically suppresses the rotational dof which does not receive stiffness contributions from any of the axia
32. es Mass Damping Harmonic and Run Analysis lines model Periodic load Choose analysis type Shear deformation Dynamic model Stiffnes mass and damping Loading Dynamic analysis frequency domain Figure 46 Analysis ribbon for forced vibration analysis in the time domain analysis arrow is active but in order to carry out an analysis the user needs to visit the Harmonic and Periodic load button which launches the dialog box in figure 47a There are three load options in the frequency domain 1 One harmonic load combination where all contributing spatial load cases have a harmonic variation with the same load frequency However the various contri buting load cases may have different phase angles This is the program s default choice see figure 47a The results from this analysis are the following steady state results for all nodal dis placements and element section forces amplitude values of dynamic amplification or dynamic load factors DLF and phase angles 2 One spatial load combination with harmonic variation applied for a series of fre quencies see figure 47b The user needs to specify a spatial load combination a frequency range and the number of frequencies within the range for which steady state response of all specified response parameters are computed The results are for each specified response parameter frequency plots of the amplitude values the dynamic load factors DLF and the phase angles 3 One periodic but n
33. eters are available and for buckling and free vibration analyses buckling factors vibration frequencies and corresponding modes are deter mined The program consists of two distinct parts a graphical user interface GUI and a computational engine Frame2D program fap2bD GUI Frame2D structural model computational model The GUI is programmed in C and OpenGL whereas Frame2D is coded in Fortran the top level subroutines in Fortran 90 and some lower level library type subroutines in Fortran 77 The user is concerned with the structural model only she he can also influence the computational model but only via the structural model The user cannot access the computational model directly except for results Units SI units are used consistently throughout the program Platform fap2d is developed and tested on Microsoft s Windows 7 It should be emphasized that the current version is a beta version Page 6 The structural model The structural model which consists of structural members interconnected at joints is referred to a global reference coordinate system x zZ see figure 1 The location of the origin of the reference coordinates is defined arbitrarily by the user member joint member joint Figure 1 Structural model A member can also be connected to a point located at the interior of an existing member such a point is called an internal joint and its associat
34. fied for selected beam compression members With reference to the dialog box in figure 27b two types of loading may be speci fied a constant loading R applied in full at time zero A 0 and a variable loading R applied gradually from zero to full magnitude in a certain number n of equal increments The user may specify the one or the other or both in combination In the latter case some of the loading is applied at time zero and maintained constant throughout whereas the rest of the loading is applied gradually in equal increments Both types of loading are defined in terms of a spatial load combination which may already be defined or it may be defined by selecting Add edit in the appropriate drop down menu Page 29 fap2D Analysis and results Computational aspects A fairly general method of nonlinear geometric analysis referred to as a co rotated formu lation with a consistent tangent stiffness is used This theory which assumes small strains but accounts for large displacements is more general than so called 2nd order theory which is basically based on small displacements For instance as opposed to 2nd order theory the method used by fam2D will activate the hammock effect which will produce a large axial force in a simply supported beam subjected to trans verse loading only provide the supports are prevented from horizontal movements It should be emphasized that linear elastic materials
35. formation Influence line Influence line analysis Figure 514 Analysis ribbon for analysis of influence lines active but in order to carry out an analysis the user needs to define a load path and one or more response parameters must have been or be defined Since influence lines are normally used to determine the maximum response to loading moving accross the load path such loading here called a load train should also be defined prior to the actual analysis Figure 52 shows the dialog boxes launched by the Define load path and the Define load train buttons respectively The load path consists of the members on which the Load path Manage load trains Load train Load trains a Load 1 00 kN Name acts in global Z direction X direction and moves along entire load path Load cases Load Distance mm Load kN 0 00 kN Show Don t show load path on model 3000 00 30 00 5000 00 30 00 When exiting this dialog 0 00 mm enable disable the possibility to add more members to load path Lv jie by clicking them Z Figure 52 Definition of load path a and load trains b load train can travel Figure 52a suggests that the load path can be defined by mark ing the relevant members before clicking the Define load path button and then click the Add selected members to the load path button that will be active if thi
36. g Free vibration Dynamic Dynamic Influence Include Default load com Run Analysis static time domain frequency domain lines Choose analysis type Shear deformation Load combination Linear static analysis Figure 21 Analysis ribbon for linear static analysis Computational aspects The computations are straightforward and the analysis is carried out for one load combination at a time By default shear deformations are not included but as already mentioned above they are easily included by simply pushing the Include shear deformation button before the Run Analysis button If bilinear axial members that is cable and or strut members are present the model is not strictly linear In this case the program will make sure through an iteration procedure that all cable and strut mem bers carry tension or compression respectively During the iteration procedure such members may thus be removed from or inserted back into the model and the itera tion continues until no bilinear members needs to be removed inserted from one iter ation to the next Typical results The frame of figure 17 subjected to the loading shown in figure 20 is analysed The results available are shown in figure 22 The main results are diagrams of displace A fal Modelling Loading Analysis Results Settings Show all Reaction force Displacement B Parameters Section ShowResultants al Force Shear Force Elastic Component
37. he first version of the tangent stiffness matrix is based on the 1st order linear stiff ness matrix K and the geometric stiffness matrix Kg is based on the axial member forces obtained by solving the problem Kor R 7 Page 30 fap2D Analysis and results If the loading Rc is sufficiently large the matrix K Ke may be indefinit and this is taken as an indication that R is more than the structure can support The load ing is halved and the procedure starts all over again If this results in a positive definite matrix the program will iterate until the unbalanced forces are suffi ciently small and a new load increment which is half of the previous loading is applied A new tangent matrix is established and tested for positive definitness If positive definite equilibrium iterations are carried out and a new load incre ment which is half of the previous one is applied if not positive definite the pro gram backs up to the last equilibrium position and halve the load increment once again before the procedure is repeated This goes on until an equilibrium position is obtained for a loading within 1 of the smallest total load that cause an indefinit tangent stiffness matrix It follows that this loading must be smaller than the loading R originally applied 2 If the loading R does not cause the tangent stiffness matrix to become negative definite the program will carry out equilibrium iterations with full loading on the structur
38. he same displacement functions as the stiffness is derived from this leads to rotational as well as translational mass and mass coupling e diagonalized mass representation this is a combination of the other two models it leads to a diagonal element mass matrix i e both translational and Page 15 fap2D The structural model rotational mass but no mass coupling NOTE The theoretical basis for this model is not very well founded The basic modelling philosophy adopted by the program see next chapter favours the lumped mass approach and only numerical reasons seem to warrant one of the other two methods In some circumstances the lumped approach may lead to numeri cal difficulties in the solution of the free vibration eigenproblem In addition to the mass of the structural members concentrated translational and rotational mass may be introduced at joints including internal joints Damping Dynamic analysis may be carried out on the complete coupled MDOF model or on a reduced decoupled SDOF model obtained through use of a limited number of modal coordinates Regardless of model viscous damping is assumed For the complete MDOF model the available damping model is the so called Rayleigh damping in which the damping matrix C is expressed as a combination of the mass matrix M and the stiffness matrix K of the model that is C a M a K The coefficients a and a may be given explicitly as input or they may be compu
39. ht hand top gs BAe z Recent structural models New Save gt 5 Test X fap2D 4 Beam_infl fap2D corner there is not much in terms of help avail 6 Euler_col fap2D Ba o able The main design criterion has been to make Eag o the use of the program as intuitive as possible G vin through familiar icons and well designed dialog D o boxes Tooltips are provided where deemed neces fap2D Settings E Exit fap2D sary useful Figure 15 Application menu On the whole the left hand mouse button is the Page 21 fap2D Modelling structure and loading operation button apology to all left handers and the right hand button is the information button The menu line has four main choices modelling loading analysis and results The fifth choice Appearance has to do with the style and coloring of the views Modelling the structure ab D Programs Frame2D Example Modelling Loading Analysis Results Settings Select member type 2 V EY O AN E R er ee en Select material ule Start SHS 100x1 yp e A Ar cA a H Beam mem i joi f Add point Add point Manage Steel End Boundary Coupling Add joint X do a Draw member spring spring ao Hinge release mass damper meshing Material F Cross section F Member Spring Joint Boundary condition and rel Mass Damping Mesh Figure 16 The modelling ribbon How to establish a viable structural m
40. ic amplification also referred to as the dynamic load factor DLF defined as the ratio between dynamic and static response the static response needs to be solved This is accomplished by solving eqn 15 for Q 0 that is K r ir Rp iR 22 where superscript s designates static Hence Kr R gt Pp 23 and Kr R gt 24 For the jth component the static response for the most un favorable position of the static harmonic wave is S S 2 S 2 rio T Viz a rjr 25 and the dynamic amplification for component j is DLF 26 S Typical results We return to the example problem in figure 43 and we apply the point load as one harmonic load with a frequency of Q 4 Hz This is a very simple example of the load ing situation of figure 47a Using the program s default settings for computational model stiffness mass and damping the steady state results obtained for the bending moments in the beam are shown in Figure 48 amplitude values figure 48a DLF values figure 48b and phase angles figure 48c a Amplitude values A amp max value 461 8 kNm Iii oii max value 18 6 Soo c I li Phase angle values amp max value 36 9 deg Figure 48 Bending moment results for the beam in figure 43 due to harmonic P Q 4hz Page 44 fap2D Analysis and results In this case the amplitude diagram in figure 48a may due to very simi
41. ic parameters General or arbitrary that is a cross section of arbitrary shape and whose section properties A 1 etc are input to the program obtained for instance by special ized cross section programs For structural members the following rules apply a b A particular member regardless of type has the same material properties throughout A particular member regardless of type has the same cross section shape throughout and with one exception the cross section is constant within the member The exception is beam and arch members having cross sections of the parametric category In this case although the shape is the same the size can vary from one joint to the next The individual shape parameters e g height width radius etc of the cross section may vary linearly from one joint to the next NOTE An internal joint may define a cross section for its host member but only if the cross section is of the parametric category and of the same shape as that of its host member Hence it is possible to describe a completely arbitrary variation of the parametric cross sections along beam and arch members Straight beam members can be divided into two or more shorter but in all other respects ordinary members that will inherit all properties of the mother member including its loading Being ordinary members means that they can after creation be assigned new properties irrespective of the mother member
42. is Figure 26 Analysis ribbon for nonlinear static analysis active but if the user push it he she will get a message telling him her that loading needs to be specified However the user may first consider to impose some kind of geometrical imperfection default is no such imperfections The shape imperfection button launches the dialog box in figure 27a A global shape imperfection may be Load history Geometric imperfection C Include global imperfection in the shape of _ 0b ____ _ _ 1 0 n 1 buckling mode deb 1 deformed geometry rv Ad 1 0 Amar NXA Constant loading Variable loading R y Xx LCmb_c R y x LCmb_v Include local sinusoidal imperfection on selected beam members with amplitude Total load R R R delta L n where n Choose define spatial load combinations for R_c and or R_v E Constant load R_c Cl Variable load R_v Number of equal load increments n Figure 27 Dialog boxes for geometric imperfection a and load history b imposed the form of which may be in the shape of a buckling mode shape due to a specific load combination or in the shape of the static deformations due to a specified load combination In either case the user must provide the maximum amplitude in mm Instead of or in addition to the global imperfection local imperfections in the shape of half sine waves with amplitudes equal to L n may be speci
43. l elements Page 18 fap2D The computational model Elements The following elements are available i gt A psa AMM Ae AV gt gt gt i bar element beam element cable element Euler Bernoulli or Timoshenko Strut element spring elements Figure 12 Elements in fan2d Beam element a simple straight 6 degree of freedom element with constant cross section Bending axial and shear deformations are considered The latter which is based on the assumption of an average shear deformation Timoshenko theory is optional Material properties are linearly elastic for all types of analysis Axial element a simple straight 4 degree of freedom element with constant cross section that can only take axial force Bi linear stiffness characteristics may be speci fied In other words a particular axial element may take both tension and compres sion in which case it is a bar element tension only in which case it is a cable element or compression only in which case it is a strut element Spring elements both linear and rotational springs may be included in the model A spring may bea boundary spring which is a spring connected to a single degree of freedom at one end and fixed to earth at the other or a coupling spring which is a spring connecting the same dof at two nodal points e g the rotation at two nodes it should be noted that the two nodes coincide geometrically For m
44. lar phase angles along the beam be taken as an actual and most severe bending moment diagram In general however the moment amplitude values over the structure will not occur at the same time We see that the dynamic effects are quite strong the largest DLF value is 18 6 for a damping ratio of 0 02 in the first two modes Keep in mind though that the load frequency is close to the resonance frequency of 3 89 Hz For a load frequency of 3 Hz the largest DLF value for the bending moment is 2 85 The phase angle shows some discrepancies at the ends of the beam these are believed to be caused by inappropriate handling of the ratio between very small numbers Results similar to those of figure 48 are available also for displacement axial force and shear force Next we apply the same point load but this time as a series of harmonic load cases with 100 different frequencies equally spaced between 1 5 and 5 Hz This is a simple example of the loading situation of figure 47b Results will only be available for spec ified response parameters of which we have one the transverse displacement under the load Again we use the program s default settings for computational model stiff ness mass and damping and the steady state results obtained for the transverse dis placement under the load are shown in Figure 49 as plots along the frequency axis of the amplitude values figure 49a DLF values figure 49b and phase angles figure 4
45. ll have 0 Mode 3 0 02 Mode 4 0 02 Mode 5 0 02 Mode 6 0 02 Mode 7 0 02 Mode 8 0 02 modei 1 with damping ratio i 0 0200 modei 1 with damping ratio _i 0 0200 ping ping a Mode 9 0 02 mode j 2 with damping ratio g j 2 0200 mode j 2 with damping ratio j 900 Mode 10 0 02 Explicit definition based on O Explicit definition based on x nis ane a2 ae ant Damping ratio 0 0200 for mode lt All gt Figure 39 Damping models It should be emphasized that all damping models assume viscous damping For modal analysis we see that the damping ratios may be computed on the basis of Rayleigh damping and assumed damping ratios of two eigenmodes to be specified if not the first and second or they may be given explicitly for each modal coordinate as in figure 39c where the All choice has been used to assign the default value of 0 02 Next we come to the Loading button and this button must be used on the first visit to Page 36 fap2D Analysis and results this ribbon The associated dialog box is shown in figure 40 Here we need to define Manage dynamic load combination Dynamic Load Combination Dynamic load combination New dynamic load combination ez EJ Default load case Own weight Figure 40 Dynamic load combination for forced vibration in the time domain at least one dynamic load combination and this co
46. ll loading at once only Figure 29 A simple EuLer column constant loading The program comes back with the message that 746 kN is Page 31 Analysis and results fap2D within 1 of the load the column can support before it becomes unstable In other words starting load of 1000 KN clearly caused a negative definite tangent stiffness The results from the analysis are shown in figure 30 a Max bending moment Max shear force 709 kN 1005 kNm Max displ Max axial force 746 kN 1448 mm Figure 30 Results for the EULER column subjected to 1000 kN applied at once We repeat the analysis but this time we apply the loading gradually in 50 equal incre ments This time results shown in figure 31 are obtained for the full load 1000 KN Max shear force 1000 kN Max bending moment 1607 kNm Max axial force Max displ 999 kN 2954 mm Figure 31 Results for the EULER column subjected to 1000 kN applied incrementally 50 load increments Page 32 fap2D Analysis and results When comparing results it should be kept in mind that they are normalized results This is particularly important for the displacement which clearly is not drawn to scale in figures 30 and 31 If we instead of Show all click the Displacement button and then the T true displacement button in the toolbox a completely different picture appears see figure 32 And now the tensile axial forces in figure 31 make sense gt g
47. load We see that all influence lines exhibit a value of zero at the supports except the influence line for the support reaction which is equal and opposite to the point load when it is positioned at the support which make sense The Result ribbon for influence line analysis is shown in figure 54d If one or more load trains are defined the Run Xtrm Analysis arrow is active and pushing the arrow will place the load train at the most severe position for the choices made in the rib bon and a result box will give the value of the response parameter caused by the load train when in this position Figures 54 e and f show the results for the bending moment at joint 5 when the train moves from left to right e and when it moves from right to left Depending on the train these two values need not as shown by this example be the same Page 48 fap2D Analysis and results c Bending moment at joint 5 max moment 0 30 kNm aD Cy Modelling Loading Analysis Results Appearance Active influence line Active load train gt Max valu J d Extreme value analysis M bending moment at left side T1 gt mMin value Run Xtrm Analysis Influence line Load train analysis Minimum response due to T1 Result Distance from load path start 15410 00 mm Minimum response 1 61E 01 kNm Convert to load case Load case name a LCT Minimum respo
48. load combination must be specified as basis for the Ritz vectors The Stiffness and Mass buttons serve the same purpose as for free vibration analysis it is in other words possible to include geometric effects in the stiffness matrix also for this type of analysis The default settings of both buttons are reasonable in most cases The Damping button also has a default setting shown in figure 39a if an MDOF model is specified and in figure 39b if modal analysis is chosen Both may serve as reasonable first assumptions Visiting this button is therefore not a must Damping full MDOF analysis Damping modal analysis Damping modal analysis Damping type Rayleigh damping Concentrated damping only Reyleigh damping Damping matrix DC al M mass proportional C a2 K stiffness proportional C a1 M a2 K mass and stiffness proportional Implicit definition of a1 and a2 based on Damping ratios for the individual modes Based on Rayleigh damping Specified explicitly Reyleigh damping Damping matrix C a1 M mass proportional D C a2 K stiffness proportional C a1 M a2 K mass and stiffness proportional Implicit definition of a1 and a2 based on Damping ratios for the individual modes Based on Rayleigh damping Specified explicitly Damping ratio Damping ratios for participating modes Mode 1 0 02 Note Modes not listed here Mode 2 0 02 wi
49. mber of straight beam elements each with 6 degrees of freedom and constant cross section properties determined as the properties at the element s mid point Distri buted loading if present is lumped into statically equivalent concentrated loads at the nodes For the structural member in figure 10 the number of elements required is probably dictated by the member geometry However the idea of load lumping also requires a straight beam member to be subdivided into a series of shorter elements if it is sub jected to any form of distributed loading even if the geometry does not call for such subdivision For a straight member the number of computational elements is dictated by the load representation and possibly also by a varying cross section which is approximated by step wise constant section properties Page 17 fap2D The computational model This simple strategy leads to a much higher number of degrees of freedom and thus more numerical work and higher storage demands than the more conventional approach of one to one relation between member and element The simplicity of the brute force technique combined with some obvious advantages in describing curved members and geometric imperfections is believed to more than compensate for the increased computational effort and storage space It also lends itself extremely well for geometric presentation of both model and results everything boils down to simple straight lines Ref
50. mbination is made up of one or more combinations of a spatial load case and a time function If the appropriate load case and or time function is are not available at this stage the user needs to go back to the loading ribbon and define the missing items A visit to the Time axis button is also a must The corresponding dialog box is shown in figure 41 We need to give the three first parameters AA A and ax all in seconds max Time axis response station point in time at which response parameters are sampled computed and stored A time increment or step unit of time eA CHHHEOHHHOH HGH HHHH I 4 5 A z gt hs L h s AR An x maximum length of response history AA 0 000000 s A_s 0 000 S A_max 0 000 s A_R 0 000 s NOTE A_s AA and A_max MUST be given Le itv 4 Figure 41 The time axis for forced vibration in the time domain The fourth parameter Ap defines a point in the relevant time domain at which a com plete response nodal displacements element section forces and reaction forces is computed and made available for diagrammatic presentation AA is a key parameter here and we shall return to this quantity and make some comments about it in the next two sections The last button in the ribbon of figure 37 is concerned with Numerical integration Viable default values are provided and we will therefore leave it for the moment we return to it in the next sec
51. nd results Computational aspects The problem is solved by the frequency response method using as for the time domain the full coupled multi degree of freedom MDOF system or superposition of a system of decoupled single degree of freedom SDOF equations obtained by modal analysis both eigenvectors and so called load dependent Ritz vectors may be used as modal coordinates Representation of stiffness mass and damping is also the same as in the time domain The problem is to find the solution of Mi Cr Kr R Q Re 12 Q is the frequency of the applied harmonic loading We seek the particular solution of eqn 12 that is the so called steady state solution r fe 13 which has the same frequency as the loading A tilde on top of a symbol denotes a complex quantity Substituting eqn 13 into eqn 12 yields _ Q MF iQCr Kr R 14 or Kr R 15 where K K Q M iQaCc 16 is the complex dynamic stiffness matrix The solution of eqn 15 gives the complex response vector F An arbitrary response component dof number j may be expressed as _ i Qt B EEE S r Fige re 17 where Y r an F p tir 18 j JO jR JI whose real and imaginary components are rip rocosh and rj rsinB 19 Here 2 2 rig Nin thir 20 is the amplitude value of the response and Page 43 fap2D Analysis and results B atan A 21 FIR its phase angle In order to determine the dynam
52. nse due to Ti Result Distance from load path start 12604 00 mm Minimum response 1 59E 01 kNm Convert to load case 1 Load case name LCT1 e Extreme bending moment due to train T1 moving from right to left Figure 54 Influence lines and extreme value analysis Page 49 fap2D Analysis and results Steel design For structures with steel components f ap2d offers a capacity check according to the rules and regulations of Eurocode 3 for all steel members with predefined or para metric cross sections This facility is available from the result view following a linear or nonlinear static analysis Three different checks are available Parameters Section elastic stress control 1 Elastic Component i i cross section control 2 and Steel desi component control 3 1 The elastic capacity of a particular cross section is governed by initial yielding at the most stressed point of the section The VON MIsEs yield criterion is used hence yielding starts when the effective VON MISES stress is equal to the material yield strength f or rather the design strength f The code requirement is simply ot Fars 27 In 2D the effective stress in terms of axial stress o and shear stress T is 2 2 o 0 3T 28 At each node of steel members the program calculates and the capacity or utili zation index defined as K O fy 29 and presents the index as a color map
53. o a 2D world Page 50 fap2D Analysis and results assumes the structure to be secured by sufficient bracing out of plane All results apply strictly to the in plane capacity Typical results Figure 55 shows some results from steel design checks applied to a simple steel frame l 20 kN m RHS 200x100x8 3 kN m SHS 100x100x10 a grid spacing 0 5 m C1 M nox 38 4 kNm Max capacity index 0 93 ee lll O b Bending moment diagram c Elastic stress control C2 E 0 0 Max capacity index 0 82 Max capacity index 0 57 section class 1 d Section control e Component control Figure 55 Examples of steel design results following a linear static analysis Right clicking the top of the right hand column of figures 55c and d produce the result boxes of figure 56a and b respectively Page 51 fap2D Analysis and results Elastic control on member 3 Section steel design control on member 3 At element end 1 At element end 2 At element end 1 Section class 1 At element end 2 Section class 1 Combined 0 9 Combined 0 93 Combined 0 55 Combined a Elastic stress control b Section control Figure 56 Detailed steel design results apply to points C1 and C2 of figure 55 The section control formulas of Eurocode 3 have been developed for section forces obtained by linear static analysis and the component control following
54. odel is fairly straightforward again the natural mode of operation is from left to right see the modelling ribbon in figure 16 The program has predefined 4 materials Steel Concrete Timber and Aluminum all with typical parameters However the user may define hers or his own material types by selecting Add edit in the pull down menu Next all cross sections to be used in the model should be selected if of predefined category or defined if of parametric or arbitrary category When placing a joint explicitly via the Add joint button or implicitly as the start or end point of a member it should be kept in mind that the program default is to snap the joint to the closest grid point Grid spacing and snap can be controlled from the toolbox but it is also quite straightforward to change the coordinates of a joint once it has been created It should also be noted that once a specific function has been chosen for instance by pushing a button this function remains active in the pointer until a new function or the neutral pointer is chosen Figure 17 shows the structural model of a simple frame The hinge at point D decouples the rotation of the column local axes RHS 200x100x10 D 7 E grid spacing 0 5 m Figure 17 Structural model of a simple frame Page 22 fap2D Modelling structure and loading from that of the beam which is continuous over the column In order to include local coordinate axes point
55. of Science and Technology in 2006 as a combined project master thesis for two students The project which is still ongoing has so far included 8 students under my supervision These are 2006 2007 Sverre Eide Holst and Magnus Minsaas 2008 2009 Dagfinn Dale Kloven and Gunnar Stenrud Nilsen 2009 2010 Jan Kristian Dolven and 2010 2011 Fredrik Larsen Brita Arvik and Daniel Aase The program consists of two distinct parts a graphical user interface GUI and a computational engine Frame2D While the computational engine Fortran code has been my responsibility the implementation of the major part of the program the GUI C and OpenGL has been carried out by the students From the start the emphasis has been on the GUI and our ambition has been to develop a powerful but above all easy to use analysis tool suitable for both education and practical engineering work I would like to thank all the students who have participated in the project Your efforts have been impressive and it has been a pleasure to work with you all Trondheim in May 2011 Kolbein Bell kolbein bell ntnu no Page 4 Capabilities For a qualified user the short version is that fam2D may be used to determine e the static response according to both linear and nonlinear theory only geo metric nonlinearity is considered e the linearized buckling load s and the associated buckling mode shape s e the free undamped vibration characte
56. of fap2b The user also needs to specify the number of mode shapes to be determined default is 5 and where on the frequency axis the eigenvalues shall be extracted defined by the so called shift value o default is o 0 More about this below Computational aspects The numerical problem to solve is the general symmetric eigenproblem K w M q 0 8 where K is the stiffness matrix M is the mass matrix is the circular frequency of the free vibration A is the eigenvalue of the problem and q is the corresponding mode of vibration eigenvector Normally the program will determine a limited number n of the lowest eigenvalues and corresponding eigenvectors mode shapes so called eigenpairs A q that satisfy eqn 8 If a non zero shift 6 is specified the modified problem K uUM q 0 9 Page 34 fap2D Analysis and results is solved where u A o and K K OM The problem is equivalent to finding n roots of the characteristic polynomial in the vicinity of o on the frequency axis see figure 36 D A det K 4 M fo n eigenvalues roots of p A f NOTE o O shift Figure 36 Schematic presentation of the free vibration eigenvalue problem The eigenvalue problem of eqn 8 is by default solved by so called subspace itera tion However as for linearized buckling in the lower right hand corner of the Run analysis button the small arrow will laun
57. on harmonic load combination see figure 47c Here the user need to specify a spatial load combination a time function defining one period of loading plus some information used by the program when approximating the load by a FOURIER series of harmonic components Two parameters control the FOURIER series expansion the maximum number of Fourier terms default value is 75 and a tolerance parameter default value is 0 05 that control the actual num ber of terms used The user should not change these numbers before an inspec tion of the time function approximation has been made this approximation is available after the analysis has been carried out more below The user also has to provide the number of time increments in the period the program will determine the steady state response at each time increment The main results provided are the steady state response curves over one time period for all specified response parameters hence this type of analysis is not relevant unless at least one response parameter has been specified at a joint The only other piece of information available after a successful analysis is a plot of the FOUIER approximation of the time function Page 41 fap2D Harmonic and periodic loading Harmonic loading Periodic loading Harmonic loading One load frequency Several load frequencies One load frequency 4Load phase angle New harmonic load EE Frequency Q 0 00 Hz
58. ost types of analyses the springs are linear but nonlinear springs are also avail able for nonlinear static analysis see figure 13 In all cases the springs have identical characteristic in tension and compression S force or moment Requirement Vi lt V lt V3 lt V4 lt V S lt 8 lt 8 lt 8 lt 8 S4 S displacement see i Vv Vy V V or rotation eee 1 stiffness l z ai y up to 5 points may be specified Figure 13 Nonlinear spring stiffness Page 19 fap2D The computational model Solution The system matrices stiffness K mass M and load R are assembled to include only the unknown degrees of freedom in other words all specified dofs are omitted from the matrices Stiffness matrices and consistent mass matrices are stored in so called skyline storage format and the basic numerical operation of solving a system of linear algebraic equations is accomplished by direct GAUSSIAN elimination LDL through factorization and substitutions For the eigenvalue problems free vibration and linearized buckling the user can choose between subspace iteration which is the default method and a truncated algo rithm due to Lanczos The latter is by far the most efficient with regard to compu tational effort however subspace iteration is a well tested and fairly robust algorithm More computational details are given below for the individual types of analysis Page 20
59. paration of the loading into a constant R and a variable part R which is controlled by the user see figure 25a may be useful in many practical situations where certain loading is always constant e g dead load The buckling factor then indicates by how much the variable part of the loading can be increased before the structure becomes unstable which is normally the most interesting question Typical results The only results from a buckling analysis are the buckling mode shapes and the corre sponding buckling factor which is the factor by which the variable part of the loading must be multiplied in order to cause the structure to buckle in the corresponding mode shape In figure 25b is shown the 2nd buckling mode for the frame of figure 17 subjected to the loading of figure 20 which is all variable The first buckling mode is simple Euler buckling of column B D for which the buckling factor is somewhat smaller than that in figure 25b namely 31 14 Page 28 fap2D Analysis and results Nonlinear static analysis Figure 26 shows the ribbon for linearized buckling analysis The Run analysis is fa Modelling Loading Analysis Results Appearance i l A sa wonn gt yn linear Buckling Free vibration Dynamic Dynamic Influence Include Shape Load history Run Analysis time domain frequency domain lines imperfection Choose analysis type Shear deformation Geometry Loading Non linear static analys
60. pecific joints An external boundary condition consists of a suppressed degree of freedom dof whereas an internal boundary condition is a displacement release hinge at a joint The computational model which consists of only straight beam elements with constant cross sections is generated automatically from the structural model Each beam and arch member is replaced by 40 default number straight Euler Bernoulli beam elements no shear deformations or Timoshenko beam elements with both bending and shear deformations with constant cross sections For short members this number may be somewhat excessive but it can be easily adjusted for the entire model or indi vidual members Bar cable and strut members are all modelled by one beam element that can only transmit axial force The user can easily control the number of ele ments representing a beam arch member locally for individual members or globally for all members in the model All distributed loading on beam arch members is lumped into statically equivalent concentrated nodal forces Page 5 fap2D Capabilities Computed results are nodal displacements section forces M V and N for each ele ment maximum and minimum axial stress at both ends of each element for all types of cross sections and maximum shear stress for most cross sections as well as residual forces at joints reaction and hinge forces For some types of analyses x y plot of specified response param
61. ristics frequencies and mode shapes e the dynamic response due to time dependent loading in the time domain e the dynamic response due to harmonic loading in the frequency domain e the dynamic response due to periodic but non harmonic loading in the frequency domain and e influence lines for specified response parameters due to a travelling load ona specified travel path for any valid 2D frame type structure This version also includes design of steel members according to Eurocode 3 The structure is modelled by straight beam or curved arch members circular or para bolic both of which can accommodate bending moment shear and axial force and or straight bar cable tension only and strut compression only members all of which can only accommodate axial force The members are interconnected at joints Elastic springs both boundary springs and coupling springs as well as eccentricities in the form of rigid but weightless arms at member ends may also be included The spatial loading may be uniform or linearly varying distributed load on beam arch members concentrated loads including moments at joints prescribed displacements at joints and initial strain e g temperature The time variation of spatial loading is defined by several different types of time functions Boundary conditions both external and internal may be specified at joints in global reference axes or local axes defined at s
62. s approach is used or the Define load path button can be clicked without any mem bers marked as was the case for the dialog box in figure 52a and then exit the box with enable and click the relevant members The dialog box in figure 52a also lets the user define the size and direction of the moving point load that produce the influ ence line s In order for the extreme response calculations described below to work properly the unit value of the point load should not be changed However its direc tion must coincide with that of the loads of the load train s A named load train can consist of any number of point loads at arbitrary but user defined mutual distances and any number of load trains can be defined Figure 52b Page 47 fap2D Analysis and results shows a load train named T1 that consists of three point loads the first of which has a magnitude of 20 KN whilst the other two both are 30 KN The distance between the first and the second is 2m whereas the last load follows 3m behind the second one Computational aspects Influence line analysis requires a completely linear model Hence bi linear members cables and or struts cannot be present in the structural model if present they must be removed or converted into bar members before an analysis is attempted The computations consist of a large series of linear static analyses one for each load situ ation A load situation in turn consists of the unit point load
63. splacement of the loaded mid point at each 5th time increment that is 0 005 s The duration of the response range is set equal to that of the loaded period hence max 10 0 s The last parameter Ap we leave alone and its value of zero indicates no system results With these parameters and the default choice for the other buttons i e MDOF computational model only material stiffness lumped mass model Ray leigh damping based on a damping ratio 0 02 in the two lowest natural modes and an HHT a integration scheme with the default values of figure 42 the vertical dis placement of the beam mid point as a function of time is shown in figure 45a We repeat the analysis with one change instead of the MDOF model we now use modal analysis with the 10 lowest modes of free vibration as modal coordinates Other wise the same assumptions as before The time history plot for this model is shown in figure 45b We repeat the analysis yet again still modal analysis but this time we use 10 load dependent Ritz vectors as modal coordinates The point load is used as loading for the Ritz vectors The time history plot for this model is shown in figure 45c We see that with the resolution of the plots in figure 45 it is impossible to distinguish the three plots For this simple example that is hardly surprising Another comment is that the time increment used in this case 0 001s probably is unnecessarily small This can easily be established by
64. ssume all loading to be variable or you can assume both variable and constant loading the significance of this choice is explained below but the upshot is that the computed buckling factors only apply to the variable loading Having made Load options Buckling of structure loaded by only variable loading 5 variable and constant loading Variable loading Press here buckling factor 34 31 Constant loading Note Prescribed displacements and or initial strains n only contribute to constant loading Note Computed buckling factors apply only to the variable part of the loading Loi a Load option dialog box b Typical buckling mode mode 2 Figure 25 Load options a and typical buckling mode b this choice the spatial load combination s need s to be specified The number of modes may be specified in the ribbon 5 is the default choice as may inclusion of shear deformations Computational aspects The total loading R may consist of a constant part R and a variable part pR The variable part is assumed to vary proportionally with a multiplier p that is R R pR 1 where R is the nominal part of the variable loading expressed by a spatial load com bination It should be noted that the variable part R can only contain external loading no prescribed displacements or initial strains Linearized buckling analysis is concerned with the 2nd order stiffness matrix K K
65. t a P 746KN fig 30 b P 1000kN fig 31 Figure 32 True displacements The results of figures 30 31 and 32 are of more academic than practical interest On closer inspection we find that stresses are of magnitude 15 000 MPa hence material failure will have occurred long before the displacements of these figures are attained The lesson here is that loads that can possibly vary should preferably be applied incrementally Another useful result available after a nonlinear analysis is the time history of defined response parameters Figure 33 shows how the horizontal displacement of the mid point of the column varies with time A which is really a measure of the exter nal load for the loading case of figure 31 This type of result requires a that Current response parameter u x displacement at joint 2 Figure 33 Horizontal displacement vs tilme or loading response parameters have been defined and b that the load is applied incrementally preferably with a significant number of load increments Page 33 fap2D Analysis and results Free undamped vibration analysis Figure 34 shows the ribbon for free vibration analysis The Run analysis arrow is Bt D Programs Frame2D Examples Beam Modelling Loading Analysis Results Appearance i r Wi e AS a a pi pe ay ip E number of modes 5 y Linear Non linear Buckling Free vibration Dynamic Dynamic Influence Inclu
66. ted by the program on the basis of generalized mass and stiffness more about this later The user can specify mass proportional damping a 0 stiffness proportional damping a 0 or a complete Rayleigh damping both a and a have non zero values For a complete MDOF model it is also possible to include point dampers viscous dashpots at any free non specified dof of any joint in addition to or instead of the Rayleigh damping For an SDOF model that is modal analysis damping ratios may be specified expli citly as input for each contributing mode or alternatively the damping ratios may be computed implicitly for each mode using a Rayleigh type approach more about this later For this SDOF model point dampers cannot be included Page 16 The computational model Basic philosophy Once the structural model and its spatial loading is complete one of several analyses may be specified Depending on the type of analysis some more information may be needed mostly concerning the loading before the analysis can be started more about this later As and when an analysis starts the structural model is automatically con verted into a computational model This transformation is based on the philosophy indicated by figure 10 Beam and arch members are subdivided into a fairly large load nodal point node NL element straight and with j constant properties Figure 10 Basic modelling concept nu
67. ter will accommodate the own weight of the structural model a loading that is automatically computed by the program If one or more pre scribed displacements are specified by the user these will be associated with an auto matically created LC called Prescribed displ Similarly if temperature and or any other form of initial strain is defined all such loading is accommodated by an auto Figure 7 Initial strain Page 12 fap2D The structural model matically created load case called Init strain It should be noted that a specific model can only have one LC Prescribed displ for prescribed displacements and one LC for initial strain temperature Init strain Computations are carried out for named load combinations LCmb not load cases A spatial load combination is a linear combination of any number of named LC s Each selected LC contributes by a user specified constant load factor The user may define any number of distinctly named load combinations and any one load combination can contain any number of individual LC s The program creates automatically an LCmb called Default load combination which on creation contains only the Default load case with a load factor of 1 0 If no loads have been assigned to the Default load case the Default load combination consists of the load case Own weight times 1 0 It follows from this that it will always be possible to carry out a lin ear static analysis for own weight only without h
68. tion Page 37 fap2D Analysis and results Computational aspects The equation of motion for the MDOF model Mr Cr Kr R t 10 or the decoupled equation 2 0 x R t 11 for the modal SDOF system are both solved by implicit numerical integration With out going into details NEWMARK s B method and the modified HHT a method due to HILBER HUGHES and TAYLOR are available for the integration task NEWMARK s method is governed by two parameters B and y which together with the size of the time step At define the variation of the acceleration over a time step the stability of the solution the amount of algorithmic damping and the accuracy of the method The most com monly used values are y 0 5 and B 0 25 constant average acceleration over the time step for which the method is unconditionally stable For y 0 5 and p 0 16667 the method describes linear acceleration which is only conditionally stable In the HHT o method a third parameter is introduced in addition to B and y The purpose of this method is to include algorithmic damping of high frequency noise without much loss of accuracy For 1 3 lt a lt 0 and y 1 20 2 and f 1 a 7 4 the method is unconditionally stable Default values for the HHT a method which is the default choice are shown in figure 42 Numerical integration HHT a method Newmark s B method Values a WAYA in the range 0 33 to 0
69. well with the last period of the Response parameters Current response parameter w z displacement at joint 2 a pa One period of B r r a steady state response Show approx load variation Response parameters Current response parameter w z displacement at joint 2 bi Show re 1 b d FourIER series 04 4 sa M approximation of o time function Figure 50 Periodic non harmonic analysis response a and time function approximation b time domain analysis shown in figure 45 where 10 periods of the same loading were analyzed as a time series By clicking the button in the left hand corner of the result box in figure 50a we get the visualization of the FOURIER series approximation of the time function shown in figure 50b The tolerance parameter 0 05 see figure 47c is satisfied by 44 FOURIER terms We recognize the GIBBS phenomenon at all points of abrupt change in the time function Page 46 fap2D Analysis and results Influence lines and extreme response Figure 51 shows the ribbon for analysis of influence lines The Run analysis arrow is at D Programs Fr Modelling Loading Analysis Results Appearance H 1 pl a pa S Fe i 7 ip ii ath Linear Non linear Buckling Free vibration Dynamic Dynamic 1 Include Define Define Run analysis static static time domain frequency domain load path load train Choose analysis type Shear de

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